Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings

Suppose $p\ge 1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2\pi)$,where$\dot{F}(e^{it})=\frac{$\mathrm{d}$}{$\mathrm{d}$t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\mathrm{\parallel }{w}_z{\mathrm{\parallel }}_{L^p}$ and $\mathrm{\parallel }\overline{w_{\stackrel{ˉ}{z}}}{\mathrm{\parallel }}_{L^p}$, where $1\leq p<2$\mathrm{.}Furthermore,if$w$isaharmonicquasiregularmappingof$$\mathbb{D}$$, then we show that $w_z$ and $\overline{w_{\stackrel{ˉ}{z}}}$ are in the Hardy space $H^p$, where $1\le p\le \mathrm{\infty }$. The corresponding Hardy norm estimates, $\mathrm{\parallel }{w}_z{\mathrm{\parallel }}_p$ and $\mathrm{\parallel }\overline{w_{\stackrel{ˉ}{z}}}{\mathrm{\parallel }}_p$, are also obtained.